8) Let U be a unitary matrix. Show that: a. U is normal. b. If2 is...
8) Let U be a unitary matrix. Show that: a. U is normal. b. Ifa is an eigenvalue of U then N= 1 (Hint: 1 Uxl =1 xl for every x e C”.
8) Let U be a unitary matrix. Show that: a. U is normal. b. If 2 is an eigenvalue of Uthen 2=1 (Hint: 1 Uxl l xl for every xC".
Let U E Mat(n; C) be a unitary matrix. Show that (UzUy) = (y) for any 7,7 € C", where U acts by the usual matrix multiplication and (-:-) is the standard inner product on Ch.
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
I need help with 1.b) 1a. Assume the unitary operator U-exp(1?/4lâ??? +â?â01. show that the matrix relation for the beam splitter, is identical with the unitary transformation Hint: use the Baker-Hausdorf lemma on page 13 of Gerry/Knight. 1b. operator U-exp(??/2la??, + âtaol). Obviously, for ?-?/2 the result in 1a is retrieved. Calculate the beam splitter matrix corresponding to the more general unitary
Question B 7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
Let Is A iagonalizable? Find an upper triangular matrix B and a unitary matrix P such that B- P-1AP. Let Is A iagonalizable? Find an upper triangular matrix B and a unitary matrix P such that B- P-1AP.
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian matrix. PROVE: If A is positive definite the n every eigenvalue of A is positiv e. (c) Let Abe an n X n Hermitian matrix. PROVE: If every eigenvalue of A is positive. Then A is positive definite. (a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian...
4. Let T be the time reversal operator. Show that T=U*K where U is the unitary operator and K is the Operator of conjugation . Use the relation TST --S describing time reversed spin operator S to show that T-UK where U = 1 sigmay
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of